FIG. 1 roughly outlines the principle of a time-continuous passive harmonic rejection (HR) mixer. A voltage source, νS, drives a variable resistor network to generate an output current equal to
                              i          L                =                                            g              m                        ·                                          v                s                            2                                =                      α            ⁢                                                  ⁢                                          G                m                            ·                                                v                  s                                2                                                                        (        1        )            where Gm may be regarded as a maximum gain or conductance value, and where the term α=cos(2πfLOt) provides the harmonic rejection by avoiding conversion at multiples of fLO. The α term may be regarded as a sinusoidal scaling factor operating on Gm.
Assuming the mixer illustrated in FIG. 1 has ideal termination—i.e., zero ohm termination at its input and output ports—the mixer will exhibit a sinusoidally varying transconductance given as gm=αGm. The cross-coupled structure of the mixer ensures that the input and output conductance of the mixer remains constant with respect to α because G1+G2=Gm.
In practice, the mixer terminations will not be ideal, i.e., they will not be zero ohms. (Here, it should be remembered that zero ohms equates to infinite conductance.) FIG. 2 illustrates the real-world presence of non-ideal mixer terminations, where GS denotes the non-ideal source termination conductance at the mixer input port, and GL denotes the non-ideal load termination conductance at the mixer output port.
In the non-ideal termination case, the transfer function of the mixer becomes a function of the source and load termination conductances and is given as
                              i          L                =                                            g              m                        ⁢                          v              s                                =                                    v              s                        (                                          α                ⁢                                                                  ⁢                                                      G                    m                                    /                  2                                                            1                +                                                      G                    m                    2                                                                              G                      S                                        ⁢                                          G                      L                                                                      +                                                                            G                      m                                        ⁡                                          (                                                                        G                          S                                                +                                                  G                          L                                                                    )                                                                                                  G                      S                                        ⁢                                          G                      L                                                                      -                                                                            α                      2                                        ⁢                                          G                      m                      2                                                                                                  G                      S                                        ⁢                                          G                      L                                                                                            )                                              (        2        )            However, while ideal terminations are not possible in practice, the design convention is to make the termination conductance values such that the transfer function substantially minors the ideal termination case. That is, if the load and source conductances satisfyGSGL>>Gm2  (3)then the mixer's actual transfer function will approximate Eq. (1), corresponding to the ideal case. Thus, while infinite conductance values cannot be achieved in practice, conventional mixer design dictates the use of termination conductances that essentially remove the influence of those terminations on mixer operation.
Such practice is understandable because Eq. (2) demonstrates that the lower the termination conductance the less ideal the mixer behavior. This fact is exemplified in FIG. 3, where the ratio of third harmonic (H3) from Eq. (2) to the fundamental harmonic (H1) with α=cos(2πfLOt) is shown as a function of relative termination conductances. Here, “relative” connotes the size relationship between GS or GL and Gm. In particular, FIG. 3 represents a contour plot of the third harmonic relative to the fundamental frequency as a function of the mixer's source and load termination conductances.
A harmonic rejection mixer with time-continuous operation according to the description above is not feasibly implemented. A more practical realization operates on a discrete-time basis and uses a so-called mixer “unit cell,” such as is shown in FIG. 4. The unit cell contains a pair of resistors having a resistance Ru=1/Gu.
The unit cell also includes a “sign-switching” network controlled by control signals S and Sbar, where Sbar is the logical inverse of S. One sees that the sign-switching network comprises pairs of cross-coupled switches that take on a specific open/closed pattern responsive to the values of S and Sbar. Thus, the S/Sbar signals applied to the unit cell discretely control its conductance.
In making a complete time-discrete harmonic rejection mixer, a vector of N unit cells are connected in parallel as shown in FIG. 5—where the cells are identified as “MUCs” for mixer unit cells. The switches are controlled to generate a time-discrete sinusoidal transconductance. Thus, the effective Local Oscillator (LO) waveform is represented by a thermometer code S<1:N>, S(k)={0, 1}, such that the differential transconductance of the mixer becomes (assuming ideal termination)
                              g          m                =                              G            u                    ⁢                                    ∑                              k                =                1                            N                        ⁢                          (                                                S                  ⁡                                      (                    k                    )                                                  -                                  Sbar                  ⁡                                      (                    k                    )                                                              )                                                          (        4        )            
A digital sequencer provides the S/Sbar control signals for the vector of unit cells. The sequencer may be, for example, a memory that is cyclically accessed to obtain the unit cell sign pattern sequence. The digital sequencer thus generates the sequence of control words (S/Sbar sign patterns) corresponding to the series of time-discrete sinusoidal transconductance values desired for the mixer.
Assuming mixer terminations that are ideal or sufficiently close to ideal, the discretization of the mixer into N unit cells means that the mixer transconductance will be uniformly quantized into N+1 levels given bygmε[−NGu−(N−2)Gu . . . (N−2)GuNGu].  (5)
Thus, α=cos(2πfLOt) will be quantized too, leading to more or less quantization error in dependence on the number of waveform samples used to represent one LO period and in strong dependence on the number of unit cells available in the mixer.
This latter dependence arises from the fact that the quantization function provided by the unit cells has a quantization resolution that improves directly with an increasing number of unit cells. Thus, the ability to adjust the transconductance value of the mixer to obtain the discrete transconductance points corresponding to the desired sinusoidal waveform requires a certain quantization resolution. Put simply, the number of unit cells in the mixer dictates the transconductance quantization resolution.
The downside of high unit cell counts includes, among other things, higher costs, higher power consumption, and larger circuit sizes. Somewhat mitigating these issues, there are known techniques for reducing quantization errors when using given numbers of unit cells. In particular, it is known to use an integer number of samples per LO period and, correspondingly, to use the number of unit cells that leads to the highest spectral purity—i.e., minimized harmonic content and therefore maximized harmonic rejection. More broadly, there are known techniques for finding optimal combinations of unit cell counts and phase offsets for the LO sinusoid, for specified numbers of samples per LO period.
Such approaches demonstrate that is possible to find a relatively low number of quantization levels required to accurately represent a sinusoid with a given number of samples per period, but the number of required quantization levels can still be quite high in practice. The required number is particularly high for cases with a large number of samples per LO period. As noted, increasing the unit cell count dramatically affects mixer size and the complexity of unit cell control signal distribution, so the known optimization approaches do not provide an effective path to minimizing unit cell counts.
Also, note that even with sufficient quantization resolution, high harmonic rejection, say >60 dB, requires rather high relative conductance termination. This point is made in FIG. 3. A poor termination at one mixer port needs to be compensated for by a corresponding increase in termination conductance at the other port. Furthermore, the termination conductance must be maintained over the entire range of LO harmonics for which high harmonic rejection is required. In this regard, high termination conductance becomes increasingly difficult to realize as the frequency range increases, and harmonic rejection performance is therefore prone to degrading with increasing frequency.